Daugavet centers are separably determined

A linear bounded operator $G$ acting from a Banach space $X$ into a Banach space $Y$ is a Daugavet center if every linear bounded rank-$1$ operator $Tcolon X to Y$ fulfills $|G+T|=|G|+|T|$. We prove that $G colon X to Y$ is a~Daugavet center if and only if for every separable subspaces $X_1subset X$ and $Y_1subset Y$ there exist separable subspaces $X_2subset X$ and $Y_2subset Y$ such that $X_1subset X_2$, $Y_1subset Y_2$, $G(X_2)subset Y_2$ and the restriction $G|_{X_2} colon X_2 to Y_2$ of $G$ is a Daugavet center. We apply this fact to study the set of $G$-narrow operators.